library(dplyr)
library(ggplot2)
library(tidyr)
library(ggplot2)
library(reshape2)
library(data.table)
library(broom)
library(olsrr)
library(corrplot)
library(plotly)

Import data

setwd("D:/Project")
air <- read.csv("Air.csv", sep=";", dec=",")
air <- air[,-c(16, 17)]

Make it long and check data. We can see outliers in -200 - replace it with NA. We can see that one of columns - Non Metanic HydroCarbons concentration have a lot of NAs. Let’s drop it and then drop NAs. We saved data loosing one column. Also we can notice that not all columns have normal distribution

str(air)
## 'data.frame':    9471 obs. of  15 variables:
##  $ Date         : Factor w/ 392 levels "","01/01/2005",..: 116 116 116 116 116 116 129 129 129 129 ...
##  $ Time         : Factor w/ 25 levels "","00.00.00",..: 20 21 22 23 24 25 2 3 4 5 ...
##  $ CO.GT.       : num  2.6 2 2.2 2.2 1.6 1.2 1.2 1 0.9 0.6 ...
##  $ PT08.S1.CO.  : int  1360 1292 1402 1376 1272 1197 1185 1136 1094 1010 ...
##  $ NMHC.GT.     : int  150 112 88 80 51 38 31 31 24 19 ...
##  $ C6H6.GT.     : num  11.9 9.4 9 9.2 6.5 4.7 3.6 3.3 2.3 1.7 ...
##  $ PT08.S2.NMHC.: int  1046 955 939 948 836 750 690 672 609 561 ...
##  $ NOx.GT.      : int  166 103 131 172 131 89 62 62 45 -200 ...
##  $ PT08.S3.NOx. : int  1056 1174 1140 1092 1205 1337 1462 1453 1579 1705 ...
##  $ NO2.GT.      : int  113 92 114 122 116 96 77 76 60 -200 ...
##  $ PT08.S4.NO2. : int  1692 1559 1555 1584 1490 1393 1333 1333 1276 1235 ...
##  $ PT08.S5.O3.  : int  1268 972 1074 1203 1110 949 733 730 620 501 ...
##  $ T            : num  13.6 13.3 11.9 11 11.2 11.2 11.3 10.7 10.7 10.3 ...
##  $ RH           : num  48.9 47.7 54 60 59.6 59.2 56.8 60 59.7 60.2 ...
##  $ AH           : num  0.758 0.726 0.75 0.787 0.789 ...
for (i in 3:ncol(air)) {
  air[,i][which(air[,i] == -200)] <- NA
}
air_long <- melt(air)
## Warning in melt(air): The melt generic in data.table has been passed a
## data.frame and will attempt to redirect to the relevant reshape2 method;
## please note that reshape2 is deprecated, and this redirection is now
## deprecated as well. To continue using melt methods from reshape2 while both
## libraries are attached, e.g. melt.list, you can prepend the namespace like
## reshape2::melt(air). In the next version, this warning will become an error.
air_long <- air_long[,-c(1,2)]
ggplot(air_long, aes(value)) + 
  geom_histogram() + 
  facet_wrap(~variable, scales = "free")
## Warning: Removed 18183 rows containing non-finite values (stat_bin).

air <- air[,c(1:4,6:15)]
air <- drop_na(air)
air_long <- melt(air)
## Warning in melt(air): The melt generic in data.table has been passed a
## data.frame and will attempt to redirect to the relevant reshape2 method;
## please note that reshape2 is deprecated, and this redirection is now
## deprecated as well. To continue using melt methods from reshape2 while both
## libraries are attached, e.g. melt.list, you can prepend the namespace like
## reshape2::melt(air). In the next version, this warning will become an error.
air_long <- air_long[,-c(1,2)]
summary(air)
##          Date            Time          CO.GT.        PT08.S1.CO.  
##  02/04/2005:  24   10.00.00: 312   Min.   : 0.100   Min.   : 647  
##  03/04/2005:  24   20.00.00: 310   1st Qu.: 1.100   1st Qu.: 956  
##  15/03/2005:  24   09.00.00: 309   Median : 1.900   Median :1085  
##  16/03/2005:  24   12.00.00: 309   Mean   : 2.182   Mean   :1120  
##  18/03/2005:  24   18.00.00: 309   3rd Qu.: 2.900   3rd Qu.:1254  
##  19/03/2005:  24   21.00.00: 309   Max.   :11.900   Max.   :2040  
##  (Other)   :6797   (Other) :5083                                  
##     C6H6.GT.     PT08.S2.NMHC.       NOx.GT.        PT08.S3.NOx.   
##  Min.   : 0.20   Min.   : 390.0   Min.   :   2.0   Min.   : 322.0  
##  1st Qu.: 4.90   1st Qu.: 760.0   1st Qu.: 103.0   1st Qu.: 642.0  
##  Median : 8.80   Median : 931.0   Median : 186.0   Median : 786.0  
##  Mean   :10.55   Mean   : 958.5   Mean   : 250.7   Mean   : 816.9  
##  3rd Qu.:14.60   3rd Qu.:1135.0   3rd Qu.: 335.0   3rd Qu.: 947.0  
##  Max.   :63.70   Max.   :2214.0   Max.   :1479.0   Max.   :2683.0  
##                                                                    
##     NO2.GT.       PT08.S4.NO2.   PT08.S5.O3.         T               RH       
##  Min.   :  2.0   Min.   : 551   Min.   : 221   Min.   :-1.90   Min.   : 9.20  
##  1st Qu.: 79.0   1st Qu.:1207   1st Qu.: 760   1st Qu.:11.20   1st Qu.:35.30  
##  Median :110.0   Median :1457   Median :1006   Median :16.80   Median :49.20  
##  Mean   :113.9   Mean   :1453   Mean   :1058   Mean   :17.76   Mean   :48.88  
##  3rd Qu.:142.0   3rd Qu.:1683   3rd Qu.:1322   3rd Qu.:23.70   3rd Qu.:62.20  
##  Max.   :333.0   Max.   :2775   Max.   :2523   Max.   :44.60   Max.   :88.70  
##                                                                               
##        AH        
##  Min.   :0.1847  
##  1st Qu.:0.6941  
##  Median :0.9539  
##  Mean   :0.9856  
##  3rd Qu.:1.2516  
##  Max.   :2.1806  
## 

So, we need to normolize our data.

normalize <- function(x) {
return ((x - min(x)) / (max(x) - min(x)))
}
air_norm <- as.data.frame(lapply(air[3:14], normalize))
air_long <- melt(air_norm)
## Warning in melt(air_norm): The melt generic in data.table has been passed
## a data.frame and will attempt to redirect to the relevant reshape2 method;
## please note that reshape2 is deprecated, and this redirection is now
## deprecated as well. To continue using melt methods from reshape2 while both
## libraries are attached, e.g. melt.list, you can prepend the namespace like
## reshape2::melt(air_norm). In the next version, this warning will become an
## error.
## No id variables; using all as measure variables

Check our data. Quantile - the proportion of cases that are less than certain values. If the requirements of “normality” are here - the data should lie on a diagonal line. We can see that several columns do not have normal distribution.

#CO.GT
qqnorm(air_norm$CO.GT., pch = 1, frame = FALSE)
qqline(air_norm$CO.GT., col = "steelblue", lwd = 2)

#PT08.S1.CO
qqnorm(air_norm$PT08.S1.CO., pch = 1, frame = FALSE)
qqline(air_norm$PT08.S1.CO., col = "steelblue", lwd = 2)

#C6H6.GT
qqnorm(air_norm$C6H6.GT., pch = 1, frame = FALSE)
qqline(air_norm$C6H6.GT., col = "steelblue", lwd = 2)

#PT08.S2.NMHC
qqnorm(air_norm$PT08.S2.NMHC., pch = 1, frame = FALSE)
qqline(air_norm$PT08.S2.NMHC., col = "steelblue", lwd = 2)

#NOx.GT
qqnorm(air_norm$NOx.GT., pch = 1, frame = FALSE)
qqline(air_norm$NOx.GT., col = "steelblue", lwd = 2)

#PT08.S3.NOx
qqnorm(air_norm$PT08.S3.NOx., pch = 1, frame = FALSE)
qqline(air_norm$PT08.S3.NOx., col = "steelblue", lwd = 2)

#NO2.GT
qqnorm(air_norm$NO2.GT., pch = 1, frame = FALSE)
qqline(air_norm$NO2.GT., col = "steelblue", lwd = 2)

#PT08.S4.NO2
qqnorm(air_norm$PT08.S4.NO2., pch = 1, frame = FALSE)
qqline(air_norm$PT08.S4.NO2., col = "steelblue", lwd = 2)

#PT08.S5.O3
qqnorm(air_norm$PT08.S5.O3., pch = 1, frame = FALSE)
qqline(air_norm$PT08.S5.O3., col = "steelblue", lwd = 2)

#T
qqnorm(air_norm$T, pch = 1, frame = FALSE)
qqline(air_norm$T, col = "steelblue", lwd = 2)

#RH
qqnorm(air_norm$RH, pch = 1, frame = FALSE)
qqline(air_norm$RH, col = "steelblue", lwd = 2)

#AH
qqnorm(air_norm$AH, pch = 1, frame = FALSE)
qqline(air_norm$AH, col = "steelblue", lwd = 2)

Response C6H6.GT. We can see columns that have linear response to C6H6.GT.

mod <- lm(data = air_norm, C6H6.GT. ~ CO.GT.)
summary(mod)
## 
## Call:
## lm(formula = C6H6.GT. ~ CO.GT., data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.23935 -0.02496 -0.00249  0.02357  0.76733 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.0050753  0.0009115   5.568 2.67e-08 ***
## CO.GT.      0.8952134  0.0042471 210.782  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.04321 on 6939 degrees of freedom
## Multiple R-squared:  0.8649, Adjusted R-squared:  0.8649 
## F-statistic: 4.443e+04 on 1 and 6939 DF,  p-value: < 2.2e-16
augment(mod)%>%
   ggplot(aes(x = CO.GT., y = C6H6.GT.))+
  geom_point()+
  geom_line(aes(y= .fitted), color = "blue", size = 1)

residuals(mod) %>% hist(main = "Residuals CO.GT.")

plot(mod, which = 1)

mod1 <- lm(data = air_norm, C6H6.GT. ~ PT08.S1.CO.)
summary(mod1)
## 
## Call:
## lm(formula = C6H6.GT. ~ PT08.S1.CO., data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.18826 -0.03559 -0.00288  0.03079  0.70768 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.059959   0.001613  -37.18   <2e-16 ***
## PT08.S1.CO.  0.656927   0.004312  152.37   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0564 on 6939 degrees of freedom
## Multiple R-squared:  0.7699, Adjusted R-squared:  0.7699 
## F-statistic: 2.322e+04 on 1 and 6939 DF,  p-value: < 2.2e-16
augment(mod1)%>%
   ggplot(aes(x = PT08.S1.CO., y = C6H6.GT.))+
  geom_point()+
  geom_line(aes(y= .fitted), color = "blue", size = 1)

residuals(mod1) %>% hist(main = "Residuals PT08.S1.CO.")

plot(mod1, which = 1)

mod2 <- lm(data = air_norm, C6H6.GT. ~ PT08.S2.NMHC.)
summary(mod2)
## 
## Call:
## lm(formula = C6H6.GT. ~ PT08.S2.NMHC., data = air_norm)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.018385 -0.015140 -0.007871  0.007973  0.287650 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -0.0856861  0.0006204  -138.1   <2e-16 ***
## PT08.S2.NMHC.  0.7980363  0.0018053   442.1   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.02177 on 6939 degrees of freedom
## Multiple R-squared:  0.9657, Adjusted R-squared:  0.9657 
## F-statistic: 1.954e+05 on 1 and 6939 DF,  p-value: < 2.2e-16
augment(mod2)%>%
   ggplot(aes(x = PT08.S2.NMHC., y = C6H6.GT.))+
  geom_point()+
  geom_line(aes(y= .fitted), color = "blue", size = 1)

residuals(mod2) %>% hist(main = "Residuals PT08.S2.NMHC.")

plot(mod2, which = 1)

mod3 <- lm(data = air_norm, C6H6.GT. ~ NOx.GT.)
summary(mod3)
## 
## Call:
## lm(formula = C6H6.GT. ~ NOx.GT., data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.20002 -0.06256 -0.01753  0.04394  0.61415 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.062395   0.001528   40.84   <2e-16 ***
## NOx.GT.     0.597921   0.006951   86.01   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.08179 on 6939 degrees of freedom
## Multiple R-squared:  0.516,  Adjusted R-squared:  0.5159 
## F-statistic:  7398 on 1 and 6939 DF,  p-value: < 2.2e-16
augment(mod3)%>%
   ggplot(aes(x = NOx.GT., y = C6H6.GT.))+
  geom_point()+
  geom_line(aes(y= .fitted), color = "blue", size = 1)

residuals(mod3) %>% hist(main = "Residuals NOx.GT.")

plot(mod3, which = 1)

mod4 <- lm(data = air_norm, C6H6.GT. ~ PT08.S3.NOx.)
summary(mod4)
## 
## Call:
## lm(formula = C6H6.GT. ~ PT08.S3.NOx., data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.14285 -0.05823 -0.01275  0.03632  0.69946 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.330684   0.002140  154.49   <2e-16 ***
## PT08.S3.NOx. -0.799672   0.009101  -87.87   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.08089 on 6939 degrees of freedom
## Multiple R-squared:  0.5267, Adjusted R-squared:  0.5266 
## F-statistic:  7721 on 1 and 6939 DF,  p-value: < 2.2e-16
augment(mod4)%>%
   ggplot(aes(x = PT08.S3.NOx., y = C6H6.GT.))+
  geom_point()+
  geom_line(aes(y= .fitted), color = "blue", size = 1)

residuals(mod4) %>% hist(main = "Residuals PT08.S3.NOx.")

plot(mod4, which = 1)

mod5 <- lm(data = air_norm, C6H6.GT. ~ NO2.GT.)
summary(mod5)
## 
## Call:
## lm(formula = C6H6.GT. ~ NO2.GT., data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.28123 -0.05625 -0.00966  0.04071  0.68139 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.004056   0.002881  -1.408    0.159    
## NO2.GT.      0.494451   0.007848  63.005   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.09377 on 6939 degrees of freedom
## Multiple R-squared:  0.3639, Adjusted R-squared:  0.3638 
## F-statistic:  3970 on 1 and 6939 DF,  p-value: < 2.2e-16
augment(mod5)%>%
   ggplot(aes(x = NO2.GT., y = C6H6.GT.))+
  geom_point()+
  geom_line(aes(y= .fitted), color = "blue", size = 1)

residuals(mod5) %>% hist(main = "Residuals NO2.GT.")

plot(mod5, which = 1)

mod6 <- lm(data = air_norm, C6H6.GT. ~ PT08.S4.NO2.)
summary(mod6)
## 
## Call:
## lm(formula = C6H6.GT. ~ PT08.S4.NO2., data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.15656 -0.05635 -0.00724  0.04597  0.79223 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.065500   0.002506  -26.14   <2e-16 ***
## PT08.S4.NO2.  0.563771   0.005755   97.96   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.07616 on 6939 degrees of freedom
## Multiple R-squared:  0.5803, Adjusted R-squared:  0.5803 
## F-statistic:  9596 on 1 and 6939 DF,  p-value: < 2.2e-16
augment(mod6)%>%
   ggplot(aes(x = PT08.S4.NO2., y = C6H6.GT.))+
  geom_point()+
  geom_line(aes(y= .fitted), color = "blue", size = 1)

residuals(mod6) %>% hist(main = "Residuals PT08.S4.NO2.")

plot(mod6, which = 1)

mod7 <- lm(data = air_norm,C6H6.GT.  ~ PT08.S5.O3.)
summary(mod7)
## 
## Call:
## lm(formula = C6H6.GT. ~ PT08.S5.O3., data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.24981 -0.03674 -0.00059  0.03326  0.52482 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.045328   0.001642  -27.61   <2e-16 ***
## PT08.S5.O3.  0.573303   0.004063  141.11   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.05977 on 6939 degrees of freedom
## Multiple R-squared:  0.7416, Adjusted R-squared:  0.7415 
## F-statistic: 1.991e+04 on 1 and 6939 DF,  p-value: < 2.2e-16
augment(mod7)%>%
   ggplot(aes(x = PT08.S5.O3., y = C6H6.GT.))+
  geom_point()+
  geom_line(aes(y= .fitted), color = "blue", size = 1)

residuals(mod7) %>% hist(main = "Residuals PT08.S5.O3.")

plot(mod7, which = 1)

mod8 <- lm(data = air_norm,C6H6.GT. ~ `T`)
summary(mod8)
## 
## Call:
## lm(formula = C6H6.GT. ~ T, data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.17626 -0.08503 -0.02819  0.05791  0.86597 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.113686   0.003377   33.66   <2e-16 ***
## T           0.116814   0.007286   16.03   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1155 on 6939 degrees of freedom
## Multiple R-squared:  0.03572,    Adjusted R-squared:  0.03558 
## F-statistic: 257.1 on 1 and 6939 DF,  p-value: < 2.2e-16
augment(mod8)%>%
   ggplot(aes(x = `T`, y = C6H6.GT.))+
  geom_point()+
  geom_line(aes(y= .fitted), color = "blue", size = 1)

residuals(mod8) %>% hist(main = "Residuals T")

plot(mod8, which = 1)

mod9 <- lm(data = air_norm, C6H6.GT. ~ RH)
summary(mod9)
## 
## Call:
## lm(formula = C6H6.GT. ~ RH, data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.16472 -0.08857 -0.02776  0.06256  0.83736 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.168841   0.003508  48.131   <2e-16 ***
## RH          -0.011576   0.006434  -1.799   0.0721 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1175 on 6939 degrees of freedom
## Multiple R-squared:  0.0004662,  Adjusted R-squared:  0.0003221 
## F-statistic: 3.236 on 1 and 6939 DF,  p-value: 0.07206
augment(mod9)%>%
   ggplot(aes(x = RH, y = C6H6.GT.))+
  geom_point()+
  geom_line(aes(y= .fitted), color = "blue", size = 1)

residuals(mod9) %>% hist(main = "Residuals RH")

plot(mod9, which = 1)

mod10 <- lm(data = air_norm, C6H6.GT. ~ AH)
summary(mod10)
## 
## Call:
## lm(formula = C6H6.GT. ~ AH, data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.19607 -0.08744 -0.02681  0.06081  0.86382 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.119150   0.003096   38.49   <2e-16 ***
## AH          0.109438   0.006899   15.86   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1155 on 6939 degrees of freedom
## Multiple R-squared:  0.035,  Adjusted R-squared:  0.03486 
## F-statistic: 251.6 on 1 and 6939 DF,  p-value: < 2.2e-16
augment(mod10)%>%
   ggplot(aes(x = AH, y = C6H6.GT.))+
  geom_point()+
  geom_line(aes(y= .fitted), color = "blue", size = 1)

residuals(mod10) %>% hist(main = "Residuals AH")

plot(mod10, which = 1)

pairs(air_norm[,sapply(air_norm, is.double)])

Check correlation.

corrplot.mixed(cor(air_norm, method = "kendall"), number.cex = .7)

Check multicolinearity. I decided to see what’s going on by hands. For multic. we need to have CI>30 and VIF>10. Also, we can see it when coeficients of columns alone are not significant but together they give significant p-value.

mod <-  lm(C6H6.GT.~PT08.S2.NMHC. + NOx.GT. + PT08.S3.NOx. + NO2.GT. + PT08.S4.NO2., data = air_norm)
summary(mod)
## 
## Call:
## lm(formula = C6H6.GT. ~ PT08.S2.NMHC. + NOx.GT. + PT08.S3.NOx. + 
##     NO2.GT. + PT08.S4.NO2., data = air_norm)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.107541 -0.011598 -0.003860  0.007867  0.273775 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -0.128346   0.001736 -73.928  < 2e-16 ***
## PT08.S2.NMHC.  0.811767   0.005821 139.463  < 2e-16 ***
## NOx.GT.        0.104145   0.003158  32.974  < 2e-16 ***
## PT08.S3.NOx.   0.130556   0.003545  36.827  < 2e-16 ***
## NO2.GT.       -0.042904   0.002939 -14.597  < 2e-16 ***
## PT08.S4.NO2.   0.019687   0.003667   5.369 8.19e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01863 on 6935 degrees of freedom
## Multiple R-squared:  0.9749, Adjusted R-squared:  0.9749 
## F-statistic: 5.389e+04 on 5 and 6935 DF,  p-value: < 2.2e-16
X <- model.matrix(~PT08.S2.NMHC. + NOx.GT. + PT08.S3.NOx. + NO2.GT. + PT08.S4.NO2., data = air_norm)
XX <- t(X) %*% X
eigen <- eigen(XX)
CI <- sqrt(max(eigen$values) / min(eigen$values))
CI
## [1] 38.71032
# Also can count it via formula 1/1-R^2
ols_coll_diag(mod)
## Tolerance and Variance Inflation Factor
## ---------------------------------------
##       Variables  Tolerance       VIF
## 1 PT08.S2.NMHC. 0.07043174 14.198143
## 2       NOx.GT. 0.25131144  3.979126
## 3  PT08.S3.NOx. 0.34956090  2.860732
## 4       NO2.GT. 0.28139450  3.553730
## 5  PT08.S4.NO2. 0.14737223  6.785539
## 
## 
## Eigenvalue and Condition Index
## ------------------------------
##    Eigenvalue Condition Index    intercept PT08.S2.NMHC.     NOx.GT.
## 1 5.139259035        1.000000 0.0005983670  4.445038e-04 0.002970002
## 2 0.598258522        2.930932 0.0017127027  1.080741e-03 0.053610656
## 3 0.182088746        5.312619 0.0001264032  9.057705e-03 0.130894451
## 4 0.059729372        9.275905 0.0029993206  8.942371e-06 0.506597242
## 5 0.013526138       19.492311 0.8122319948  1.333775e-01 0.028926663
## 6 0.007138187       26.832201 0.1823312118  8.560306e-01 0.277000986
##   PT08.S3.NOx.     NO2.GT. PT08.S4.NO2.
## 1  0.001760862 0.001513613 6.783985e-04
## 2  0.065813839 0.002081996 3.339913e-05
## 3  0.042990083 0.018947690 4.823296e-02
## 4  0.079221335 0.432104143 1.009362e-02
## 5  0.806637576 0.118860131 6.737641e-04
## 6  0.003576305 0.426492427 9.402879e-01
residuals(mod) %>% hist(main = "Residuals PT08.S2.NMHC. + NOx.GT. + PT08.S3.NOx. + NO2.GT. + PT08.S4.NO2.")

plot(mod, which = 1)

mod <-  lm(C6H6.GT.~NOx.GT. + PT08.S3.NOx. + NO2.GT.+PT08.S4.NO2., data = air_norm)
summary(mod)
## 
## Call:
## lm(formula = C6H6.GT. ~ NOx.GT. + PT08.S3.NOx. + NO2.GT. + PT08.S4.NO2., 
##     data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.13920 -0.02335 -0.00319  0.01927  0.61839 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.143806   0.003379 -42.557   <2e-16 ***
## NOx.GT.       0.356365   0.005050  70.564   <2e-16 ***
## PT08.S3.NOx.  0.008813   0.006702   1.315    0.189    
## NO2.GT.       0.158388   0.004994  31.717   <2e-16 ***
## PT08.S4.NO2.  0.472327   0.003329 141.894   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03634 on 6936 degrees of freedom
## Multiple R-squared:  0.9045, Adjusted R-squared:  0.9045 
## F-statistic: 1.643e+04 on 4 and 6936 DF,  p-value: < 2.2e-16
X <- model.matrix(~NOx.GT. + PT08.S3.NOx. + NO2.GT.+PT08.S4.NO2., data = air_norm)
XX <- t(X) %*% X
eigen <- eigen(XX)
CI <- sqrt(max(eigen$values) / min(eigen$values))
CI
## [1] 21.5376
# Also can count it via formula 1/1-R^2
ols_coll_diag(mod)
## Tolerance and Variance Inflation Factor
## ---------------------------------------
##      Variables Tolerance      VIF
## 1      NOx.GT. 0.3739089 2.674448
## 2 PT08.S3.NOx. 0.3721233 2.687282
## 3      NO2.GT. 0.3708137 2.696772
## 4 PT08.S4.NO2. 0.6803347 1.469865
## 
## 
## Eigenvalue and Condition Index
## ------------------------------
##   Eigenvalue Condition Index    intercept     NOx.GT. PT08.S3.NOx.     NO2.GT.
## 1 4.21432517        1.000000 9.087272e-04 0.006362944  0.002997369 0.002945636
## 2 0.56258305        2.736974 1.141258e-03 0.113454921  0.066527093 0.005957732
## 3 0.15130645        5.277586 8.538575e-05 0.123219339  0.081423501 0.012714553
## 4 0.05972556        8.400089 2.918017e-03 0.755643783  0.082724495 0.568361415
## 5 0.01205978       18.693662 9.949466e-01 0.001319013  0.766327542 0.410020664
##   PT08.S4.NO2.
## 1 4.556101e-03
## 2 1.072926e-05
## 3 4.090600e-01
## 4 4.973504e-02
## 5 5.366381e-01
residuals(mod) %>% hist(main = "Residuals NOx.GT. + PT08.S3.NOx. + NO2.GT.+PT08.S4.NO2.")

plot(mod, which = 1)

mod <-  lm(C6H6.GT.~NOx.GT. + PT08.S3.NOx.+PT08.S4.NO2., data = air_norm)
summary(mod)
## 
## Call:
## lm(formula = C6H6.GT. ~ NOx.GT. + PT08.S3.NOx. + PT08.S4.NO2., 
##     data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.15046 -0.02558 -0.00366  0.02252  0.62349 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.079501   0.002893  -27.48   <2e-16 ***
## NOx.GT.       0.446413   0.004469   99.89   <2e-16 ***
## PT08.S3.NOx. -0.067482   0.006693  -10.08   <2e-16 ***
## PT08.S4.NO2.  0.447808   0.003464  129.26   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03888 on 6937 degrees of freedom
## Multiple R-squared:  0.8907, Adjusted R-squared:  0.8906 
## F-statistic: 1.884e+04 on 3 and 6937 DF,  p-value: < 2.2e-16
X <- model.matrix(~ NOx.GT.  + PT08.S3.NOx.+PT08.S4.NO2., data = air_norm)
XX <- t(X) %*% X
eigen <- eigen(XX)
CI <- sqrt(max(eigen$values) / min(eigen$values))
CI
## [1] 19.31043
# Also can count it via formula 1/1-R^2
ols_coll_diag(mod)
## Tolerance and Variance Inflation Factor
## ---------------------------------------
##      Variables Tolerance      VIF
## 1      NOx.GT. 0.5466887 1.829194
## 2 PT08.S3.NOx. 0.4271601 2.341043
## 3 PT08.S4.NO2. 0.7191206 1.390587
## 
## 
## Eigenvalue and Condition Index
## ------------------------------
##   Eigenvalue Condition Index    intercept    NOx.GT. PT08.S3.NOx. PT08.S4.NO2.
## 1 3.31230936        1.000000 2.323060e-03 0.01406454  0.005966555  0.007892873
## 2 0.52303786        2.516511 7.982151e-04 0.23453090  0.072741803  0.001422487
## 3 0.14668190        4.752008 5.012680e-06 0.33438471  0.105029622  0.396339407
## 4 0.01797088       13.576280 9.968737e-01 0.41701984  0.816262020  0.594345233
residuals(mod) %>% hist(main = "Residuals NOx.GT. + PT08.S3.NOx. + PT08.S4.NO2")

plot(mod, which = 1)

mod <-  lm(C6H6.GT.~NOx.GT. +PT08.S4.NO2., data = air_norm)
summary(mod)
## 
## Call:
## lm(formula = C6H6.GT. ~ NOx.GT. + PT08.S4.NO2., data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.16081 -0.02592 -0.00350  0.02360  0.62284 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.105500   0.001320  -79.91   <2e-16 ***
## NOx.GT.       0.475674   0.003423  138.97   <2e-16 ***
## PT08.S4.NO2.  0.464895   0.003043  152.76   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03916 on 6938 degrees of freedom
## Multiple R-squared:  0.8891, Adjusted R-squared:  0.889 
## F-statistic: 2.781e+04 on 2 and 6938 DF,  p-value: < 2.2e-16
X <- model.matrix(~NOx.GT. +PT08.S4.NO2., data = air_norm)
XX <- t(X) %*% X
eigen <- eigen(XX)
CI <- sqrt(max(eigen$values) / min(eigen$values))
CI
## [1] 8.491566
# Also can count it via formula 1/1-R^2
ols_coll_diag(mod)
## Tolerance and Variance Inflation Factor
## ---------------------------------------
##      Variables Tolerance      VIF
## 1      NOx.GT.  0.945341 1.057819
## 2 PT08.S4.NO2.  0.945341 1.057819
## 
## 
## Eigenvalue and Condition Index
## ------------------------------
##   Eigenvalue Condition Index  intercept      NOx.GT. PT08.S4.NO2.
## 1 2.64611815        1.000000 0.01668626 0.0446849114   0.01658563
## 2 0.28497117        3.047222 0.06870324 0.9551867442   0.06551725
## 3 0.06891068        6.196713 0.91461051 0.0001283444   0.91789712
residuals(mod) %>% hist(main = "Residuals NOx.GT. +PT08.S4.NO2.")

plot(mod, which = 1)

Final model Residuals should be normally distributed and the Q-Q Plot will show this. If residuals follow close to a straight line on this plot, it is a good indication they are normally distributed.

# Good model but NOx.GT. not normally distributed
# I've decided not to transform data, because I don't want to see log-log transformation
mod <-  lm(C6H6.GT.~NOx.GT. + PT08.S3.NOx. + NO2.GT.+PT08.S4.NO2., data = air_norm)
summary(mod)
## 
## Call:
## lm(formula = C6H6.GT. ~ NOx.GT. + PT08.S3.NOx. + NO2.GT. + PT08.S4.NO2., 
##     data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.13920 -0.02335 -0.00319  0.01927  0.61839 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.143806   0.003379 -42.557   <2e-16 ***
## NOx.GT.       0.356365   0.005050  70.564   <2e-16 ***
## PT08.S3.NOx.  0.008813   0.006702   1.315    0.189    
## NO2.GT.       0.158388   0.004994  31.717   <2e-16 ***
## PT08.S4.NO2.  0.472327   0.003329 141.894   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03634 on 6936 degrees of freedom
## Multiple R-squared:  0.9045, Adjusted R-squared:  0.9045 
## F-statistic: 1.643e+04 on 4 and 6936 DF,  p-value: < 2.2e-16
X <- model.matrix(~NOx.GT. + PT08.S3.NOx. + NO2.GT.+PT08.S4.NO2., data = air_norm)
XX <- t(X) %*% X
eigen <- eigen(XX)
CI <- sqrt(max(eigen$values) / min(eigen$values))
CI
## [1] 21.5376
# Also can count it via formula 1/1-R^2
ols_coll_diag(mod)
## Tolerance and Variance Inflation Factor
## ---------------------------------------
##      Variables Tolerance      VIF
## 1      NOx.GT. 0.3739089 2.674448
## 2 PT08.S3.NOx. 0.3721233 2.687282
## 3      NO2.GT. 0.3708137 2.696772
## 4 PT08.S4.NO2. 0.6803347 1.469865
## 
## 
## Eigenvalue and Condition Index
## ------------------------------
##   Eigenvalue Condition Index    intercept     NOx.GT. PT08.S3.NOx.     NO2.GT.
## 1 4.21432517        1.000000 9.087272e-04 0.006362944  0.002997369 0.002945636
## 2 0.56258305        2.736974 1.141258e-03 0.113454921  0.066527093 0.005957732
## 3 0.15130645        5.277586 8.538575e-05 0.123219339  0.081423501 0.012714553
## 4 0.05972556        8.400089 2.918017e-03 0.755643783  0.082724495 0.568361415
## 5 0.01205978       18.693662 9.949466e-01 0.001319013  0.766327542 0.410020664
##   PT08.S4.NO2.
## 1 4.556101e-03
## 2 1.072926e-05
## 3 4.090600e-01
## 4 4.973504e-02
## 5 5.366381e-01
ols_plot_resid_fit_spread(mod)

ols_plot_obs_fit(mod)

mod <-  lm(C6H6.GT.~CO.GT. + PT08.S4.NO2., data = air_norm)
summary(mod)
## 
## Call:
## lm(formula = C6H6.GT. ~ CO.GT. + PT08.S4.NO2., data = air_norm)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.16444 -0.01879 -0.00216  0.01556  0.76399 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.051025   0.001126  -45.33   <2e-16 ***
## CO.GT.        0.718581   0.004320  166.32   <2e-16 ***
## PT08.S4.NO2.  0.215265   0.003322   64.81   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03411 on 6938 degrees of freedom
## Multiple R-squared:  0.9159, Adjusted R-squared:  0.9158 
## F-statistic: 3.776e+04 on 2 and 6938 DF,  p-value: < 2.2e-16
X <- model.matrix(~CO.GT. + PT08.S4.NO2., data = air_norm)
XX <- t(X) %*% X
eigen <- eigen(XX)
CI <- sqrt(max(eigen$values) / min(eigen$values))
CI
## [1] 13.33846
ols_coll_diag(mod)
## Tolerance and Variance Inflation Factor
## ---------------------------------------
##      Variables Tolerance      VIF
## 1       CO.GT. 0.6020484 1.660996
## 2 PT08.S4.NO2. 0.6020484 1.660996
## 
## 
## Eigenvalue and Condition Index
## ------------------------------
##   Eigenvalue Condition Index  intercept     CO.GT. PT08.S4.NO2.
## 1 2.76742714        1.000000 0.01578747 0.02263258  0.010088266
## 2 0.18068900        3.913562 0.29140318 0.63351920  0.006558737
## 3 0.05188386        7.303347 0.69280934 0.34384822  0.983352997
ols_plot_resid_fit_spread(mod)

ols_plot_obs_fit(mod)

Separated data in 75:25% We can see that R^2 is close to 1 which says about linear dependence. P-value is close to 0 - independent variables explain the dynamics of the dependent variable.

set.seed(2)
sep <- sample.int(n = nrow(air_norm), size = floor(.75*nrow(air_norm)))
train <- air_norm[sep,]
test <- air_norm[-sep,]
train1 <- train[,c('C6H6.GT.', 'CO.GT.', 'PT08.S4.NO2.')]
test1 <- test[,c('C6H6.GT.', 'CO.GT.', 'PT08.S4.NO2.')]
mod1 <- lm(C6H6.GT.~CO.GT. + PT08.S4.NO2., data = train1)
a1 <- summary(mod1)
pred1 <- predict(mod1, newdata = test1)
test1$C6H6.GT._pred <- pred1
head(test1)
test1 <- rbindlist(list(test1[,c(2,3,1)], test1[,c(2,3,4)]))
train1$type <- 'train'
test1$type <- 'test'
test1[1:(nrow(test1)/2),4] <- 'real value'
all <- rbind(train1, test1)
plot_ly(data = all,
        z = ~C6H6.GT.,
        y = ~PT08.S4.NO2.,
        x = ~CO.GT., opacity = 0.7, color = ~type)%>%
  layout(title = paste("R2", round(a1$r.squared, 3),
                                  sep = ": "),
                            paste("pvalue", 2.2e-16, sep = ": "),
         xaxis = list(title = "CO.GT.",
                      zeroline = FALSE),
         yaxis = list(title = "PT08.S4.NO2.",
                      zeroline = FALSE),
         zaxis = list(title = "C6H6.GT.",
                      zeroline = FALSE))